Integrated Attitude Determination and Control System via Magnetic Measurements and Actuation

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation Integrated Attitude Determination and Control System via Magnetic Measurements and Actuation MOHAMMAD ABDELRAHMAN AND SANG-YOUNG PARK Astrodynamics and Control Lab (ACL), Department of Astronomy and Space Science Yonsei University -79, Seoul, Republic of KOREA Um_abdelrahman@galaxy.yonsei.ac.krU, Uspark@galaxy.yonsei.ac.krU Abstract: - A nonlinear control scheme using a Modified State Dependent Riccati Equation MSDRE has been developed through a pseudo-linearization of spacecraft augmented dynamics and kinematics. The full-state knowledge, required for the control loop is provided through a generalized algorithm for spacecraft three-axis attitude and rate estimation based on the utilization of magnetometer measurements and their time derivatives, while the control torque is generated via magnetorquers. The resulted attitude determination and control system has shown the capability of estimating the attitude better than deg and rate of order. deg/sec in addition to maintain the pointing accuracy within deg in each axis with pointing stability of less than. deg/sec. Key-Words: - Attitude Control, Attitude Estimation, Magnetometer, Magnetorquers, Modified Riccati Equation, Nonlinear Control. Introduction The aim of this paper is to develop a low cost three-axis Attitude Determination and Control System ADCS to fulfill spacecraft mission requirements during low accuracy modes which are the most frequent modes during mission life time. The current work introduces a generalized nonlinear control scheme using a Modified State Dependent Riccati Equation MSDRE. The system dynamics equation is represented by the spacecraft nonlinear dynamics with momentum bias including gravity gradient, aerodynamic, and magnetic residual torques. Then, the quaternion kinematics is augmented with spacecraft dynamics to represent the overall process dynamics. A pseudo-linear formulation of the augmented system is developed while a MSDRE controller derived to solve a trajectory tracking/model following problem. The derivatives of the state dependent matrix w.r.t the states are taken into account through the development of the final MSDRE controller such that an optimal control signal is obtained rather than producing a suboptimal controller when these derivates are ignored. To obtain the optimum control signal full-state knowledge is required, so attitude and rate filter is integrated to the control loop. The proposed filter is previously developed by the authors for spacecraft three-axis attitude and rate estimation utilizing magnetometer measurements only [7]. The global asymptotic stability of the controller is investigated by applying Lyapunov theorem, and concluded by introducing stability regimes for the overall system which verify the stability conditions of the controller and the required accuracy of the filter. To test the developed ADCS, EgyptSat-, launched in the last April, is used as a real test case. Basically, the ADCS has been tested to control the satellite and estimate the attitude and rates during detumbling and standby modes where the magnetometer measurements and magnetorquers only can fulfill the mission attitude and rate accuracy requirements for these two modes. The next section describes briefly the statement of the problem then the MSDRE controller is developed and adopted for tracking problem. Through the fourth section the pseudo-linear representation of the system is derived and the stability issues are introduced. Before the simulation section, the state estimator design appears and discussed. Finally, the paper is concluded by testing the proposed ADCS via simulations. Statement of the Problem Consider the nonlinear system xx = ff(xx, tt) νν xx (tt) () we will try to control this system such that its output is expressed as yy = cccc () with the availability of a set of measurements zz = h(xx, tt) νν zz (tt) () where νν xx (tt) and νν zz (tt) are random zero-mean processes described by the process and measurement noise matrices QQ ff = EE(νν xx νν xx TT ) and RR ff = EE(νν zz νν zz TT ). Now the task is to design a controller such that the system tracks a specified reference trajectory while a state estimator shall be plugged in to provide the necessary states to drive the controller. The reference trajectory is given by χχ = FFFF () yy rrrrrr = CCCC () the pair [FF, CC] is supposed to be completely observable. ISSN: 797 6 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation The MSDRE Controller Design The control approach will be based on using the Differential State Dependent Riccati Euation DSDRE to solve tracking/model following problem. Let s start by converting the nonlinear dynamics into a pseudo-linear system xx = AA(xx)xx GGGG (6) where uu is the control signal. Note the process noise part is dropped in the controller design and it will be included through the estimator development. A new dynamic system can be derived by augmenting the pseudo-linear system with the reference trajectory model such that xx = AA (xx, tt)xx GGuu (7) where xx = xx χχ, AA = AA FF and GG = GG. Then a quadratic cost function JJ is defined as tt ff JJ = xxtt QQxx uu TT RRRRdddd (8) tt QQ QQQQ where QQ = CC TT QQ CC TT and nonnegative definite QQQQ symmetric, while RR is positive definite symmetric. From the above, a generalized Riccati equation can be derived. PP = PPAA AA TTPP PPGGRR GG TT PP QQ AA TT xx PP GG TT uu PP PPGGRR GG TT xx ii xx ii uu (9) PP points calculated offline and can be interpolated with time so, let s try to avoid the pair[ff, CC] in the Riccati equations. Define a new vector bb eeeeee, where eeeeee stands for an external control signal as we will see Augmented System Controller xx = AA(xx)xx GGGG χχ = FFFF Fig. Augmented System Control [] bb eeeeee = PP χχ (7) bb eeeeee = PP χχ PP χχ (8) Using reference trajectory model in Eqs () and () leads to, bb eeeeee = AA TT TT xx PPPPRR GG TT bb The first part of this equation is the well known statedependent Riccati equation, and the second part is due to Similar treatment can be applied for PP and PP xx eeeeee QQyy rrrrrr (9) ii the nonlinearity of the system. Neglecting the second part defining two new vectors cc eeeeee and dd eeeeee as follows leads to only a suboptimal control. Through this work the cc eeeeee = PP TT χχ () matrix GG is only function of time, so the modified statedependent Riccati equation MSDRE for optimal control is dd eeeeee = χχ TT PP χχ () and PP = PPAA AA TTPP PPGGRR GG TT PP AA TT Then the differential equations of cc eeeeee and dd eeeeee can be xx PP QQ () written as xx ii cc eeeeee = [AA GGRR GG TT PP] TT cc eeeeee QQyy rrrrrr () with a boundary condition, PPtt ff =. This gives and immediately the optimal control uu uu = RR GG TT dd eeeeee = cc TT eeeeee GGRR GG TT bb eeeeee yy TT rrrrrr QQyy rrrrrr () PPxx () Finally the optimal control can be rewritten as The matrix PP can be partitioned as, PP = PP PP uu = KKKK uu eeeeee () () where PP PP uu this gives the optimal control as follows eeeeee = RR GG TT bb eeeeee () uu = KKKK KK χχ χχ () represents the external control signal Fig.. To summarize, we need to integrate the modified Riccati where KK = RR GG TT PP () equations for PP and bb eeeeee, then the optimal control signal and can be obtained. The boundary conditions for the KK χχ = RR GG TT PP () MSDRE are PPtt ff = and bb eeeeee tt ff =. Figure shows the augmented system controlled by use xx yy RR of linear state-variable feedback, as for a standard GG TT bb eeeeee xx = AA(xx)xx GGGG cc regulator. The modified Riccati equation becomes PP = PPPP AA TT PP PPPPRR GG TT PP TT (6) KK xx PP QQ xx ii Fig. Optimal Trajectory Control [] The reference trajectory yy rrrrrr is usually given in a form of KK χχ KK χχ xx cc yy ISSN: 797 7 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation Spacecraft Attitude Maneuver Using MSDRE This section provides a background about the spacecraft dynamics and kinematics modelling with proposed disturbances, and then a pseudo-linear representation for this model is adopted such that the application of the MSDRE scheme can be implemented.. Spacecraft Attitude Dynamics The attitude dynamics of a rigid spacecraft can be expressed by the well-known Euler s equation in presence of momentum exchange devices and given ωω = II [ωω (IIII h)] II TT h (6) The torque TT is the total torque vector exerted on the spacecraft which includes gravity gradient torque TT gggg = μμ RR SS RR SS IIRR SS (7) where RR SS is the position vector, and μμ is the Earth s gravitational constant. Then the aerodynamic torque nn TT aaaaaaaa = CC ddρρ VV rrrrrr NN ii VV rrrrrr VV rrrrrr rr ii AA ii (8) ii= The summation is taken over nn surfaces of the spacecraft with area AA ii and unit outward normal vector NN ii. The parameter CC dd is the drag coefficient and ρρ is the atmosphere density. The velocity vector relative to the rotating atmosphere is VV rrrrrr = VV SS ωω RR SS, where ωω is the Earth s rotational velocity, and VV SS is the spacecraft velocity. Also, the magnetic disturbance torque TT mm = mm BB (9) The spacecraft dipole moment is the vector mm while BB is the Earth s magnetic field. The control strategy is based on using only magnetic torquers in a momentum biased system, so the control torque is given by TT cc = MM BB () where MM is the dipole moment generated by the magnetic torquers. Let s assume that there is another control signal uu which constructs an orthogonal set with the two vectors MM, and BB, then [BB ][BB ] TT cc = BB uu = GG uu () where the anti-symmetric matrix [BB ] is defined by bb bb [BB ] = bb bb () bb bb Finally, the total torque is TT = TT gggg TT aaaaaaaa TT mm TT cc. Note all vectors in the previous equations are derived in the spacecraft body frame.. Quaternion Kinematics The spacecraft attitude is represented through the quaternion, defined by qq qq qq () while the vector part qq [qq qq qq ] TT = nn sin(αα ) and the scalar part qq = cos(αα ). Here nn is a unit vector corresponding to the principal axis of rotation and αα is the angle of rotation. The quaternion kinematics is derived through the spacecraft s angular velocity as follows qq = Ω(ωω)qq = Ξ(qq)ωω () where Ω(ωω) and Ξ(qq) are defined as [ωω ] ωω qq UU [qq ] Ω(ωω) = and Ξ(qq) =. ωω TT TT qq Here UU is a unit matrix. The quaternion four elements satisfy the following normalization constraint qq TT qq = qq TT qq qq =.. Pseudo-Linear Modelling Now the task is to derive a pseudo-linear system in the form of Eq.(6) such that the MSDRE control scheme can be applied. Starting with the state vector xx resulting from the augmentation of the spacecraft dynamics with the quaternion kinematics as follows xx TT = [ωω qq] TT () The state vector xx describes the spacecraft angular velocity and attitude in the body frame w.r.t the inertial frame.let s consider the first part of the attitude dynamic equation FF = II [ωω (IIII h)] = FF ωω II [h ]ωω (6) The attitude matrix is given by Π = (qq qq TT qq ) UU qq qq TT qq [qq ] (7) and its derivates w.r.t qq ii are Π = qq qq ii UU [qq uu TT ii uu ii qq ii TT ] qq [uu ii ] (8) uu ii is i th column of the unit matrix UU and ii =. Π = qq qq UU [qq ] (9) From the above it can be shown that Π = Π i= qq ii. If there is a vector expressed in the inertial frame TT II, its transformation to body frame TT bb is obtained by TT bb = ΠTT II = Π qq qq ii TT II = TT bb qq () i= ii qq Applying this result to the second part of the attitude dynamic equation i.e. torque vectors, leads to TT gggg = TT gggg qq, TT mm = TT mm qq, and TT aaaaaaaa = TT aaaaaaaa qq. Note the Earth s rotation effect is neglected w.r.t spacecraft velocity. Also, the quaternion kinematics can be rewritten as follows: qq = qq qq ωω qq () Now the augmented attitude dynamics and kinematics equations can be expressed in pseudo-linear form qq ii ISSN: 797 8 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation FF ωω = TT gggg TT mm TT aaaaaaaa ωω qq qq qq qq II GG uu [h ] II TT gggg TT aaaaaaaa ωω qq or xx = AA(xx)xx GGGG Γ(xx)xx () The effect of the matrix Γ(xx) on the system is negligible compared to AA(xx) and is proved by simulation as will be presented in the section of simulations results. Consider the augmented system without the control vector xx = ff (xx, tt) = AA(xx)xx () simply, AA(xx) = ff ff, where is the Jacobian xx = ff = (xx) xx AA(xx) xx ii xx ii xx ii ii () where AA(xx) ii is i th column of the matrix AA(xx). From the above (xx) xx = AA(xx). The reference trajectory can be xx ii obtained by ωω rrrrrr = ωω rrrrrr Π rrrrrr ωω rrrr () qq rrrrrr = qq rrrrrr qq oooo (6) where bb, oo, and II stand for body, orbit, and inertial frames, respectively. Usually the reference trajectory is given in body frame w.r.t orbit frame so the previous relations are used to derive the corresponding reference trajectory in TT body frame w.r.t inertial frame. Then yy rrrrrr = [ωω rrrrrr qq rrrrrr ] TT. Now, we can apply the MSDRE control scheme to spacecraft attitude tracking using the backward solution of PP = PPPP AA TT PP PPPPRR GG TT PP QQ (7) bb eeeeee = [AA TT PPPPRR GG TT ]bb eeeeee QQyy rrrrrr (8) The control gain KK, the external control component uu eeeeee, and the optimal control signal uu are obtained through Eq.(), Eq.(), and Eq.(), respectively.. Effect of Actuator Saturation Due to the actuator saturation the optimal dipole moment MM resulting from the MSDRE controller should be scaled down keeping its direction as it is. From Fig. the following relation can be obtained MM ss = MM mmmmmm MM MM (9) ii where MM ss is the scaled dipole moment, MM mmmmmm is the maximum dipole moment of the magnetic torquer, and MM ii is the maximum projected dipole moment on ii axis. M* M*s Mmax Fig. Schematic Diagram for Scaled Dipole Moment. Stability of the MSDRE Controller Based on Lyapunov s method, stability regimes of the MSDRE can be estimated. Choose VV(xx) = xx TT PPxx as a Lyapunov function candidate, where PP is the positive definite solution of the modified Riccati equation Eq.(). Rewriting VV = xx TT PPPP xx TT bb eeeeee cc TT eeeeee xx dd eeeeee () then the time derivative becomes VV = xx TT PPPP xx TT PP xx xx TT PPxx xx TT bb eeeeee xx TT bb eeeeee () cc eeeeee TT xx cc TT eeeeee xx dd eeeeee sub., with Eq.(6) where uu = RR GG TT PPPP RR GG TT bb eeeeee and using the modified Riccati equations VV = ηη TT RRRRλλ TT RR λλ xx TT QQQQ yy TT rrrrrr QQyy rrrrrr [λλ TT ηη xx TT AA TT (PPPP bb eeeeee )] () yy TT rrrrrr QQQQ xx TT QQyy rrrrrr where ηη = RR GG TT PPPP and λλ = GG TT bb eeeeee. Since QQ and RR are positive definite matrices then the first part is negative, also the last part is small and can be neglected specially TT when yy rrrrrr = [ 6 ] TT and the last term of the QQ matrix is chosen to be relatively small. Then regions should be estimated to ensure negative values of the second term and hence the stability of the system. To estimate the stability regions of the candidate MSDRE controller, uniformly random distributed initial conditions Θ and Ω on the intervals ±Θ dddddd and ±Ω dddddd/ss for each axis, are generated and the following conditions are investigated: VV(xx) > aaaaaa VV (xx) < () State Estimator Design The well known extended Kalman filter EKF is used through this work as a state estimator to provide the MSDRE controller with the necessary states required to drive the controller.. Filter Process Model The spacecraft dynamics and kinematics, given in Eqs. (6) and () are augmented to represent the system process model. Also, the spacecraft residual dipole and the drag coefficient are added to the state vector for more filter robustness. M*i ISSN: 797 9 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation ωω II [ωω (IIII h)] II TT h qq νν ωω = Ξ(qq)ωω mm νν () mm CC dd νν cc xx = ff(xx, tt) νν xx (tt) The state vector xx is eleven-dimensional, defined as xx TT = [ωω qq mm CC dd ] TT (). Filter Measurement Model The measurement equation, required for the application of the extended Kalman filtering, is derived from vector kinematics and considered to be a nonlinear function of the states according to BB bb ΠBB II BB = νν BB bb ΠBB II ωω (ΠBB II ) νν (6) BB zz = h(xx, tt) νν zz (tt) where BB II is a vector of some reference such as the position vector to the Sun, to a star or the Earth s magnetic field vector in a reference coordinate system, and BB bb is the corresponding measured vector in the spacecraft body frame. Only magnetometer measurements and its time derivatives are considered through this work. The prediction of the state estimates and covariance are accomplished by tt xx kk = ff(xx, ττ) dddd (7) PP ff = Φ kk kk PP ff Φ TT kk kk QQ ff kk (8) where Φ UU tt is the state transition matrix and PP ff is the error covariance matrix. It obviously can be shown that the derivatives needed for the Jacobian are already included in the state-dependent matrix AA(xx). Only it is needed to calculate the derivative of the control torque w.r.t the states which simply can be given as TT cc = GGGG. Note an accurate integration method should be used to perform the state propagation. The discrete process covariance QQ ff is computed according to QQ kk ff = kk tt Φ(ττ) QQ ff Φ TT (ττ)dddd. A closed form solution can be obtained as nn qq jj qq iiii ff = qq iiff tt( aa ff aa iiii tt iiii tt) jj = qq iiii ff = qq ii ff aa jjjj tt aa tt iiii (9) qq jj ff aa iiii tt nn aa tt jjjj qq ll ff aa iiiiaa jjjj tt ll= ll ii ll jj where ii, jj = NN, and nn = NN ii. Here NN is the no. of states, qq ii ff and qq iiii ff are the elements of matrices QQ ff and QQ ff respectively (QQ kk ff is assumed to be diagonal) while aa iiii is the elements of the Jacobian matrix. Finally, as with the basic discrete Kalman filter, the measurement update equations correct the state and covariance estimates with the measurements zz kk as follows KK ff = PP kk ff HH TT kk kk HH kk PP ff HH TT kk kk RR ff (6) kk xx kk = xx kk KK ff [zz kk kk h(xx kk )] (6) PP ff = UU KK kk ff HH kk kk PP ff UU KK kk ff HH kk kk TT (6) TT KK ff RR kk ff KK kk ff kk where HH = h(xx,tt) and KK ff is the Kaman gain. xx=xx kk 6 Simulation Results To test the developed ADCS, EgyptSat-, injected in sun-synchronous low Earth orbit is used as a real test case. The initial control strategy of EgyptSat- was to utilize only the magnetorquers to detumble the spacecraft after separation and then to use the magnetometer reading to estimate the attitude and rates during the standby/stabilization mode. Unfortunately, during the commissioning phase the reaction wheels had been used in additions to magnetorquers to detumble the spacecraft in order to maintain the power profile of the spacecraft, since it was found that using only magnetorquers would not detumble the spacecraft in the proper time. Also, the fiber gyros onboard had to be used for rate measurements and a strap down scheme was used for obtaining the attitude. The proposed controller/estimator design is utilized through both detumbling and standby/stabilization modes using only magnetorquers and magnetometer readings to meet the ADCS requirements, summarized by depressing the total angular velocity after separation to less than. dddddd/ssssss in one orbit and half. Then to bring the spacecraft to stabilization (Earth pointing) mode in no more than three orbits due to the spacecraft power profile. Also, to maintain the pointing accuracy during stabilization mode within dddddd and rate less than. dddddd/ssssss in each axis. Finally, to achieve attitude determination accuracy better than dddddd and rate of. dddddd/ssssss during the same mode. Before the discussion of the simulations results, the initial conditions obtained from the Launch Vehicle provider according the worst condition of separation are represented by angular velocities ωω = [ ] TT dddddd/ss, attitude in Euler Angles EEEE = [ ] TT dddddd, pitch wheel bias h = [. ] TT NN. mm. ss, position RR SS = [ 966. 68] TT kkkk, and velocity VV SS = [ 7.6.8.77] TT kkkk/ss. For the MSDRE controller, two sets of weighting matrices RR and QQ are used for the both flight modes, while the switching occurs according to value of the angular velocity and ISSN: 797 6 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation scheduled such that RR = [ 8 8 8 ] TT and QQ = dddddddd [. 8. 8. 8 ] TT [ 6 6 6.] TT for detumbling mode while for stabilization mode RR = [ 8 8 8 ] TT [. and QQ = dddddddd 8. 9. 8 ] TT [. 6. 6. 6.] TT. The switching condition is represented byωω bbbb xx. dddddd/ss and ωω bbbb zz. dddddd/ss. The three-axis magnetometer (TAM) senor noise has a standard deviation σσ BB = nnnn. The induced noise due to measurement differentiation can be calculated as νν BB = νν BB kk νν BBkk. So, it is also a zero mean white noise kk tt with a standard deviation σσ BB = σσ BB. The sampling time tt of the ADCS control loop is larger than the sampling time of the magnetometer which is considered a fast sensor. So, during one sampling interval, several magnetometer readings can be obtained. Then, the time derivatives of the magnetic field measurements are computed in a semi analytical method via differentiating the polynomials result from a cubic spline fit of the measurement data during each sampling interval. Care must be taken to use a reasonable number of readings during the sampling interval since using a low number, results in poor presentation of the magnetic field and a high number results in very noisy derivatives. To preserve the quaternion normalization constraint, different methods handled this problem such as adding a pseudo measurement or truncating the state vector. For simplicity and reduction of the computational burden, quaternion normalization; namely qq kk kk = qq kk kk qq kk kk is used through this work. The filter is initialized using initial states xx = [. ] TT and [.9.9.9 ] TT PP ff = dddddddd [ ] TT [... ] TT. as initial error covariance while, the process and measurement noise covariance matrices QQ ff and RR ff are assembled according to QQ ff = [(6. 7 ) (.99 7 ) (7. 7 ) ] TT dddddddd RR ff [ ] TT [( ) ( ) ( ) ] TT. [(. = dddddddd 8 ) (. 8 ) (. 8 ) ] TT [(6. 9 ) (6. 9 ) (6. 9 ) ] TT The rest of the simulation parameters are defined to be magnetorquer strength MM mmmmmm = [ ] TT AA. mm, spacecraft dipole mm = [...] TT AA. mm, drag coefficient CC dd =, time Span of orbit, and sampling time of ss. The integration algorithm used for the modified Riccati equation and state propagation is Runge - Kutta ode and the magnetic field vector BB is based on IGRF Model. Figure indicates the minor influence of the matrix Γ(xx) in Eq. () compared to AA(xx) and is since the resulting r.m.s error of the angular velocity is less than. dddddd/ss and almost zero for the quaternion. The total angular velocity is depressed after separation from 6 dddddd/ssssss to.7 dddddd/ssssss in. orbit and kept to less than. dddddd/ssssss upto orbits as shown in Fig.. The pointing and angular velocity errors are shown in Figs. 6 and 7 through two time windows. The first one represents the detumbling mode and extends upto orbit and the other shows the performance during the standby mode. The MSDRE controller successfully drives the spacecraft from detumbling mode to become in an Earth pointing mode in less than two orbits since the pointing error starts to be bounded within dddddddd and the angular velocity error reaches values less than. dddddd/ssssss. Then these conditions are maintained during the standby mode as shown in the right hand side of Figs. 6 and 7. Figure 8 gives a picture of the history of the applied dipole moments, and the corresponding control torque appears in Fig.9. It should be mentioned here that the hardware configuration and sizing are kept to the original values as in EgyptSat-, since the magnetorquers in addition to the reaction wheels are utilized to control the spacecraft while in the current work only magnetorquers are used. However, larger values of the dipole moments might be used to avoid reaching the saturation limits specially, during the standby mode. The stability condition in Eq. () is verified as in Fig. which indicates the history of the Lyapunov function and its time derivatives through all over the both modes. Figures through describe the performance of the proposed filter. The first two figures show the fast convergence of the filter since it successfully estimated the attitude and rate within the required values. It clearly can be shown the synchronization of the convergence of the filter and the controller, since the filter also, starts to reach the required accuracy for the attitude and rate in less than two orbits and kept this accuracy throughout the standby mode to be better than ddddgggg and less than. dddddd/ssssss for the attitude and rate respectively. Figures and indicate the estimation errors for the spacecraft dipole and the drag coefficient. Both parameters start to converge almost after 6 orbits to reach steady state values of. AA. mm and. for the spacecraft dipole and the drag coefficient respectively. Finally to estimate the stability regions of the candidate MSDRE controller, uniformly random distributed initial conditions on the intervals [ ] ddddgg and [ ] dddddd/ss for each axis are generated and stability condition in Eq. () is investigated also, the attitude and rate estimation accuracy are investigated throughout the same intervals to demonstrate filter convergence and ISSN: 797 6 ISBN: 978-96-7-8-6

Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation stability. Figures and 6 give the resulting stability regimes for the overall ADCS. 7 Conclusions A modified state-dependent Riccati equation MSDRE based controller has been developed and adopted for spacecraft attitude maneuver/tracking. An extended Kalman filter EKF based estimator has been developed for attitude and rate estimation using only one reference sensor. While the magnetometer measurements are used through this paper, any other reference sensor can be used or added to the measurement model. The stability of the controller and estimator is demonstrated by identifying the regimes of different initial conditions which lead to Lyapunov function convergence and bounded attitude and rate estimation accuracy. Acknowledgments This work is supported by the Korean Science and Engineering Foundation (KOSEF) through the National Research Laboratory Program funded by the Ministry of Science and Technology (No. M68-6j- 8). References: [] Lefferts, F. Markley, and M. Shuster, Kalman Filtering for Spacecraft Attitude Estimation, Journal of Guidance, Control, and Dynamics, Vol., No., Sept.-Oct. 98, pp. 7-9. [] B. Anderson, and J. Moore, Optimal Control, Prentice-Hall, Inc., A Division of Simon & Schuster, Englewood Cliffs, NJ 76, 989. [] R. Wisniewski, Linear Time Varying Approach to Satellite Attitude Control Using Only Electromagnetic Actuation, Journal of Guidance, Control, and Dynamics, Vol., No.,, pp. 6 67. [] M. L. Psiaki, Magnetic Torquer Attitude Control via Asymptotic Periodic Linear Quadratic Regulation, Journal of Guidance, Control, and Dynamics, Vol., No.,, pp. 86 9. [] J. R. Cloutier, and D. T. Stansbery, The Capabilities and Art of State-Dependent Riccati Equation-Based Design, Proceedings of the American Control Conference, Anchorage, Alaska, May. [6] W. Luo and Y. C. Chu, Attitude Control Using the SDRE Technique, Seventh International Conference on Control, Automation, Robotics and Vision, Singapore, December. [7] M. Abdelrahman, and S-Y Park Simultaneous Spacecraft Attitude and Estimation Using Magnetic Field Vector Measurements," Proceedings of the7 th ESA Conference on Guidance, Navigation & Control Systems, June 8, Tralee, Ireland.. W 6.. Roll (deg) Pitch (deg) Yaw (deg).8.7.6..... Fig. Total Angular Velocity History (Detumbling Transient/Standby) - - - - Fig. 6 Pointing Error (Detumbling/Transient-Standby) Fig. 7 Angular Velocity Error (Detumbling/Transient-Standby) M x (A.m ) W x W y W z - - -. -.. -. - -.. - W e RMS..8.6. M y (A.m ) M z (A.m ) -. - Fig Effect of Γ Matrix on the Spacecraft Dynamics Fig. 8 Applied Dipole Moments ISSN: 797 6 ISBN: 978-96-7-8-6

- 6 - - - 6 - Proceedings of the th WSEAS International Conference on Automatic Control, Modelling and Simulation T x (N.m) T y (N.m) T z (N.m) x - - x - - x - - Fig. 9 Applied Control Torque m y e (A.m ) m x e (A.m ) m z e (A.m ) - -. -. - Fig. Spacecraft Dipole Estimation Error. x 7 V dv/dt 8. 6 Lyapunov Function. C d e Roll e (deg) Pitch e (deg) Yaw e (deg) -. - Fig. Lyapunov Function and Its Time Derivative - Fig. attitude Estimation Error (Detumbling/Transient-Standby) W x e W y e W z e - - - - - - Fig. Angular Velocity Estimation Error (Detumbling/Transient-Standby) -.. -. -.. -.. -. Yaw (deg) W z - - - - - - - - 6 Fig. Drag Coefficient Estimation Error - - 6 - - - 6 - - 6 - - - -6 - - - - - - - - Fig. Euler Angles Stability Regimes -6 - - - - - - - - - - - -6 6 - - - - - - - - - 6 - - - - - - - -6-6 - 6 - - Pitch (deg) - 6 6 - - Roll (deg) W y - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - W x Fig. 6 Angular Velocity Stability Regimes ISSN: 797 6 ISBN: 978-96-7-8-6